Splitting polytopes

A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder). This generalizes a...

Verfasser: Herrmann, Sven
Joswig, Michael
FB/Einrichtung:FB 10: Mathematik und Informatik
Dokumenttypen:Artikel
Medientypen:Text
Erscheinungsdatum:2008
Publikation in MIAMI:30.11.2008
Datum der letzten Änderung:15.04.2015
Quelle:Münster Journal of Mathematics, 1 (2008), S. 109-142
Angaben zur Ausgabe:[Electronic ed.]
Fachgebiet (DDC):510: Mathematik
Lizenz:InC 1.0
Sprache:English
Format:PDF-Dokument
URN:urn:nbn:de:hbz:6-43529463487
Permalink:https://nbn-resolving.de/urn:nbn:de:hbz:6-43529463487
Onlinezugriff:mjm_vol_1_05.pdf

A split of a polytope P is a (regular) subdivision with exactly two maximal cells. It turns out that each weight function on the vertices of P admits a unique decomposition as a linear combination of weight functions corresponding to the splits of P (with a split prime remainder). This generalizes a result of Bandelt and Dress [Adv. Math. 92 (1992)] on the decomposition of finite metric spaces. Introducing the concept of compatibility of splits gives rise to a finite simplicial complex associated with any polytope P, the split complex of P. Complete descriptions of the split complexes of all hypersimplices are obtained. Moreover, it is shown that these complexes arise as subcomplexes of the tropical (pre-)Grassmannians of Speyer and Sturmfels [Adv. Geom. 4 (2004)].